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Factors of 64152
By Helen Latimer on January 23, 1998:

The number 64152 is equal to the product 8×9×1×891. By considering the factors of 64152, find another way of expressing 64152 as the product of three 1-digit and a 3-digit number formed from these digits.

Please could you give an explanation of how you found your answer.

from Helen Latimer(13), Nicky Luescher(13) and Elisabeth Johnson(13)

By Anonymous:

Firstly factorise 891.

This can be expressed in its simplest form as 3×3×3×3×11

The full set of primes is therefore
2×2×2
3×3×3×3×3×3
11

Firstly, we can dismiss one because only 981 would be big enough, and that does not work.

We know the three digit number must have a factor of 11 in it.

How are the 3's divided up?

If all were used in the single digit numbers, then we would have 999 which does not work

If five were used, then we would have 699 or 399 (and other combinations). Check the smallest, 399, and obtain 96957 which is > 64152
therefore there are at most four 3's in the single digit numbers and a factor of 9 in the three digit number


This means that the only possiblities for the three digit number are 9*11* {2,3,4,6,8,9)

198, 297, 396, 594, 792, 891 are the only possibilities

By trial and error, 396 works (can dismiss 891 and 198 because they contain 1's)

Sorry if this answer doesn't have the elegance you were hoping for - doubtless there are other ways of approaching the problem which might result in a quicker solution.